In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph $G$ and a positive integer $k$, and the objective is to decide whether $G$ contains a minimal vertex cover of size at least $k$. Motivated by the kernelization of MMVC with parameter $k$, our main contribution is to introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex optimization problems, which we call lop-kernels. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on $H$-free graphs for several graphs $H$, such as the bull, the paw, or the complete graphs, by making use of the Erd\H{o}s-Hajnal property in order to find an appropriate decomposition. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$. This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC.
翻译:在最大 MMDVC 问题中, 我们得到一个图形 $G$ 和正整数 $k$, 目标是决定$G$是否包含一个最小的顶层覆盖率至少为$k$。 受 MMVC 以参数 $k$ 的内脏驱动, 我们的主要贡献是引入一个简单的一般性框架, 以某种类型的多式内核度获得较低的界限, 我们称之为顶端优化问题 。 非正式地, 需要这种型内核来保存大型最佳的 Out- 内核内核解决方案, 并捕捉文献中的绝大多数现有内核内核。 作为这个框架的结果, 我们的主要贡献是引入一个简单的通用框架, 以某类多式内核内核内核的内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内